Combinatorial Identities For Stirling Numbers: The Unpublished Notes Of H W Gould

by Jocelyn Quaintance, Henry W Gould

Overview

This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Gould's techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities.This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics.

Product Details

ISBN-13:	9789814725262
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	12/14/2015
Pages:	276
Product dimensions:	6.20(w) x 9.00(h) x 0.80(d)

Table of Contents

Foreword vii

Preface ix

Acknowledgments xi

1 Basic Properties of Series 1

1.1 General Considerations of Σⁿ_k=af(k) 4

1.2 Pascal's Identity in Evaluation of Series 8

2 The Binomial Theorem 13

2.1 Newton's Binomial Theorem and the Geometric Series 16

3 Iterative Series 21

3.1 Two Summation Interchange Formulas 23

3.2 Gould's Convolution Formula 28

4 Two of Professor Gould's Favorite Algebraic Techniques 35

4.1 Coefficient Comparison 35

4.2 The Fundamental Theorem of Algebra 42

5 Vandermonde Convolution 49

5.1 Five Basic Applications of the Vandermonde Convolution 50

5.2 An In-Depth Investigation Involving Equation (5.4) 55

6 The n^th Difference Operator and Euler's Finite Difference Theorem 63

6.1 Eider's Finite Difference Theorem 68

6.2 Applications of Equation (6.16) 69

7 Melzak's Formula 79

7.1 Basic Applications of Melzak's Formula 83

7.2 Two Advanced Applications of Melzak's Formula 86

7.3 Partial Inaction Generalizations of Equation (7.1) 89

7.4 Lagrange Interpolation Theorem 93

8 Generalized Derivative Formulas 101

8.1 Leibniz Rule 101

8.2 Generalized Chain Rule 102

8.3 Five Applications of Hoppe's Formula 105

9 Stirling Numbers of the Second Kind S(n, k) 113

9.1 Euler's Formula for S(n, k) 118

9.2 GrunerCs Operational Formula 126

9.3 Expansions of <$$$> 131

9.4 Bell Numbers 133

10 Eulerian Numbers 139

10.1 Functional Expansions Involving Eulerian Numbers 142

10.2 Combinatorial Interpretation of A(n, m) 144

11 Worpitzky Numbers 147

11.1 Polynomial Expansions from Nielsen's Formula 152

11.2 Nielsen's Expansion with Taylor's Theorem 156

11.3 Nielsen Numbers 161

12 Stirling Numbers of the First Kind s(n,k) 165

12.1 Properties of s(n, k) 168

12.2 Orthogonality Relationships for Stirling Numbers 170

12.3 Functional Expansions Involving s(n,k) 173

12.4 Derivative Expansions Involving s(n,k) 175

13 Explicit Formulas for s(n,n - k) 177

13.1 Sehläfli's Formula s(n, n - k) 180

13.2 Proof of Equation (13.2) 185

14 Number Theoretic Definitions of Stirling Numbers 191

14.1 Relationships Between S₁(n,k) and S₂(n,k) 196

14.2 Hagen Recurrences for S₁(n,k) and S₂(n,k) 199

15 Bernoulli Numbers 203

15.1 Sum of Powers of Numbers 205

15.2 Other Representations of S_p(n) 212

15.3 Euler Polynomials and Euler Numbers 217

15.1 Polynomial Expansions Involving B_n(x) 223

Appendix A Newton-Gregory Expansions 227

Appendix B Generalized Bernoulli and Euler Polynomials 231

B.1 Basic Properties of B^(a)_k(x) and E^(a)_k(x) 232

B.2 Generalized Bernoulli and Euler Polynomial Derivative Expansions 239

B.3 Additional Considerations Involving Newton Series 247

Bibliography 253

Index 257